CONTROL SYSTEMS, ROBOTICS AND AUTOMATION - Vol. XII - Feedback Linearization of Nonlinear Systems - Alberto Isidori and Claudio De Persis

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1 FEEDBACK LINEARIZATION OF NONLINEAR SYSTEMS Alberto Isidori Dipartimeto di Iformatica e Sistemistica, Uiversità di Roma La Sapieza" ad Departmet of Systems Sciece ad Mathematics, Washigto Uiversity i St. Louis, Italy. Claudio De Persis Dipartimeto di Iformatica e Sistemistica, Uiversità di Roma La Sapieza", Italy. Keywords: Feedback Liearizatio, Chage of Coordiates, State Feedback. Cotets 1. The problem of feedback liearizatio 2. Normal forms of sigle-iput sigle-output systems 3. Coditios for exact liearizatio via feedback Glossary Bibliography Biographical Sketches Summary The chapter examies the feedback liearizatio problem. Relyig o the cocept of relative degree, a chage of coordiates ad a feedback law are foud for which the closed-loop system i the ew coordiates is i a ormal form. This is the poit of departure for obtaiig a costructive procedure allowig us to trasform a oliear system ito a liear ad cotrollable oe. 1. The Problem of Feedback Liearizatio Cosider a sigle-iput sigle-output oliear system modeled by differetial equatios of the type x = f( x) + g( x) u y = h( x), (1) where x is the state vector, u is the cotrol iput, y is the measured output ad f ( x), g( x), h( x) are smooth fuctios of x. A basic issue i cotrol theory is how to use feedback i order to modify the origial iteral dyamics of a cotrolled plat i such a way as to obtai the same behavior of some prescribed autoomous liear system. This problem, which i the case of liear systems is kow as the problem of pole placemet, is kow i the more geeral framework of oliear systems as feedback liearizatio (see Bibliography).

2 Chages i the descriptio ad i the behavior of system (1) will be cosidered uder two types of trasformatio: (i) chages of coordiates i the state space ad (ii) static state feedback cotrol laws, i.e. memoryless state feedback laws. I the case of a liear system, x = Ax + Bu y = Cx (2) a static state feedback cotrol law takes the form u = Fx+ Gv, (3) i which v represets a ew cotrol iput ad F ad G are matrices of appropriate dimesios. Moreover, oly liear chages of coordiates are usually cosidered. This correspods to the substitutio of the origial state vector x with a ew vector z related to x by a trasformatio of the form z = Tx, where T is a osigular matrix. Accordigly, the origial descriptio of system (2) is replaced by a ew descriptio = Az + Bu y = Cz i which 1 1 A = TAT, B = TB, C = CT. I the case of a oliear system, a static feedback cotrol law is a cotrol law of the form u =α ( x) + β( x) v, (5) where v represets a ew cotrol iput ad β( x ) is assumed to be ozero for all x. Moreover, oliear chages of coordiates are cosidered, i.e., trasformatios of the form (4) z = ( x), Φ (6) where z is the ew state vector ad Φ ( x) represets a ( -vector valued) fuctio of variables,

3 φ1( x) φ 1( x1, x 2,, x ) φ2( x) φ 2( x1, x 2,, x ) Φ ( x) = =, φ ( x) φ ( x, x,, x ) 1 2 with the followig properties: 1. Φ ( x) is ivertible; i.e. there exists a fuctio 1 1 Φ ( Φ( x)) = x, Φ( Φ ( z)) = z for all x ad all z. 2. Φ ( x) ad Φ 1 ( z) are both smooth mappigs. Φ 1 ( z) such that A trasformatio of this type is called a global diffeomorphism. The first property is eeded to guaratee the ivertibility of the trasformatio to yield the origial state vector as x = Φ 1 ( z), while the secod oe guaratees that the descriptio of the system i the ew coordiates is still a smooth oe. Sometimes a trasformatio possessig both of these properties ad defied for all x is hard to fid ad the properties i questio are difficult to check. Thus, i most cases, trasformatios defied oly i the eighborhood of a give poit are of iterest. A trasformatio of this type is called a local diffeomorphism. To check whether or ot a give trasformatio is a local diffeomorphism, the followig result is very useful. Propositio 1. Suppose Φ ( x) is a smooth fuctio defied o some ope subset U. Suppose the Jacobia matrix φ1 φ1 φ1 x1 x2 x φ2 φ2 φ 2 Φ = x1 x2 x x φ φ φ x1 x2 x 0 is osigular at poit x = x. The, for some suitable ope subset U 0 of U, 0 cotaiig x, Φ ( x) defies a local diffeomorphism betwee U 0 ad its image

4 0 Φ ( U ). The effect of a chage of coordiates o the descriptio of a oliear system ca be aalyzed as follows. Set zt () = Φ ( xt ()) ad differetiate both sides with respect to time to yield dz Φ dx Φ zt () = = = ( f( xt ()) + gxt ( ()) ut ()). dt x dt x The, expressig x( t ) as zt ( ) = f ( zt ( )) + gzt ( ( )) ut ( ) yt () = hzt ( ()), where Φ 1 ( zt ( )), oe obtais f Φ Φ ( z) = f( x), g ( z) = g( x), x 1 1 x ( z) x = Φ x= Φ ( z) hz ( ) = ( ). ( hx) 1 x= Φ ( z) The latter are the formulas relatig the ew descriptio of the system to the origial oe. Give the oliear system (1), the problem of feedback liearizatio cosists of fidig, if possible, a chage of coordiates of the form (6) ad a static state feedback of the form (5) such that the composed dyamics of (1) ad (5), amely the system (7) x = f( x) + g( x) α ( x) + g( x) β ( x) v, expressed i the ew coordiates z, is the liear ad cotrollable system = z = z = z = v.

5 - - - TO ACCESS ALL THE 21 PAGES OF THIS CHAPTER, Click here Bibliography Hut L., Su R., Meyer G. (1983). Desig for multi-iput systems. I R. Brockett, R. Millma, H. Sussma, eds., Differetial Geometric Cotrol Theory, pp , Birkhauser. Isidori A. (1995). Noliear Cotrol Systems. Spriger-Verlag. Jakubczyk B., Respodek W. (1980). O liearizatio of cotrol systems. Bull. Acad. Poloaise Sci. Ser. Sci. Math 28, Su R. (1982). O the liear equivalets of oliear systems. Systems & Cotrol Lett. 2, Biographical Sketches Alberto Isidori was bor i Rapallo, Italy, i His research iterests are primarily focused o mathematical cotrol theory ad cotrol egieerig. He graduated i electrical egieerig from the Uiversity of Rome i Sice 1975, he has bee Professor of Automatic Cotrol i this Uiversity. Sice 1989, he also holds a part-time positio of Professor of Systems Sciece ad Math. at Washigto Uiversity i St. Louis. He has held visitig positios at various academic/research istitutios which iclude: Uiversity of Florida, Gaiesville (November 1974), Washigto Uiversity, St. Louis (August- October 1980, August-December 1983), Uiversity of Califoria, Davis (July-August 1983), Arizoa State Uiversity, Tempe (August-December 1986, April-May 1989), Uiversity of Illiois, Urbaa (April-May 1987), CINVESTAV, Mexico City (September 1987), Uiversity of Califoria, Berkeley (Jauary 1988), CNRS, Paris (May 1988), ETH, Zurich (April-May 1991), Uiversite Paris-Dauphie, Paris (May 1992), NASA, Lagley (November 1996, February 1997). He is the author of several books: Teoria dei Sistemi (i Italia), with A.Ruberti, 1979; Sistemi di Cotrollo (i Italia), 1979 ad 1992; Noliear Cotrol Systems (Spriger Verlag), 1985, 1989 ad 1995; Topics i Cotrol Theory (Birkhauser), with H.Kobloch ad D.Flockerzi, 1993; Output Regulatio of Ucertai Noliear Systems (Birkhauser), with C.I. Byres ad F. Delli Priscoli, He is also editor/coeditor of ie volumes of Coferece proceedigs ad author of over 130 articles, for a large part o the subject of oliear feedback desig. He received the G.S.Axelby Outstadig Paper Award from the Cotrol Systems Society of IEEE i 1981, for his techical cotributios to the applicatio of differetial geometry to the problem of oiteractig cotrol of oliear systems, ad i 1990, for his techical cotributios to the solutio of the problem of asymptotic regulatio ad trackig i oliear systems. He also received from the IFAC the Automatica Prize i 1991 for his techical cotributios to the applicatio the otio of zero dyamics i problems of feedback stabilizatio. I 1987 he was elected Fellow member of the IEEE "for fudametal cotributios to oliear cotrol theory". I 1996, at the opeig of 13th IFAC World Cogress i Sa Fracisco, Dr. Isidori received the "Giorgio Quazza Medal". This medal is the highest techical award give by the Iteratioal Federatio of Automatic Cotrol, ad is preseted oce every third year for lifetime cotributios to automatic cotrol sciece ad egieerig. The "Giorgio Quazza Medal" was awarded to Dr. Isidori for "pioeerig ad fudametal cotributios to the theory of oliear feedback cotrol". He has orgaized or co-orgaized several iteratioal Cofereces o the subject feedback desig for oliear systems. I particular, he was the iitiator a permaet series of IFAC Symposia o this topic.

6 He is presetly servig i umerous Editorial Boards of major archival jourals, which iclude Automatica, IEEE Trasactios o Automatic Cotrol, Iteratioal Joural of Cotrol, Joural of Mathematical Systems Estimatio ad Cotrol, Iteratioal Joural of Robust ad Noliear Cotrol. He has also served i the program committee of several major iteratioal Cofereces. He acted as Program director, i the area of Systems ad Cotrol, for the Italia Departmet of Educatio from 1983 to From 1993 to 1996 he served i the Coucil of IFAC. Claudio De Persis received his Laurea degree summa cum laude i Electrical Egieerig ad his doctoral degree i Systems Egieerig i 1996 ad, respectively, 2000 both from Uiversità di Roma "La Sapieza", Rome, Italy. He held visitig positios i the Departmet of Mathematics ad Statistics, Texas Tech Uiversity, Lubbock, TX, ad i the Departmet of Mathematics, Uiversity of Califoria, Davis, CA i From November 1999 to Jue 2001 he has bee a Research Associate i the Departmet of Systems Sciece ad Mathematics, Washigto Uiversity i St. Louis, MO. Sice July 2001 he has bee a Postdoctoral Research Associate i the Departmet of Electrical Egieerig, Yale Uiversity, New Have, CT. O November 1, 2002, he took up his ew positio as Assistat Professor i the Departmet of Computer ad Systems Sciece "A. Ruberti", Uiversità di Roma "La Sapieza". He has give cotributios to the theory of fault detectio for oliear systems, switched systems ad supervisory cotrol with costraits. His curret research iterests iclude observatio ad cotrol with limited iformatio, hybrid systems, moitorig i large-scale systems, complex systems, etworks, moder commuicatio, post-geomic biology.